Final answer:
The perimeter of a triangle formed by three given points can be found using the distance formula. By determining the equation for the perimeter of triangle AOP, we can find the locus of point P that satisfies the given condition. The correct locus equation is option B.
Step-by-step explanation:
To find the locus of point P such that the perimeter of triangle AOP is 4, we need to determine all the points that satisfy this condition. The perimeter of a triangle is the sum of the lengths of its sides. The given points are O(0,0) and A(0,1), and we need to find the locus of P(x,y) such that the perimeter of triangle AOP is 4.
The distance between two points (x1, y1) and (x2, y2) is given by the distance formula: sqrt((x2-x1)^2 + (y2-y1)^2). So the perimeter of triangle AOP is given by: OA + OP + PA. We know that OA = 1 and PA = sqrt(x^2 + (y-1)^2). So we can write the equation for the perimeter as:
Perimeter = 1 + sqrt(x^2 + (y-1)^2) + sqrt(x^2 + y^2) = 4.
We can then simplify and manipulate this equation to find the locus of P. The correct option is B.